Non-Markovianity of a quantum emitter in front of a mirror

We consider a quantum emitter ("atom") radiating in a one-dimensional (1D) photonic waveguide in the presence of a single mirror, resulting in a delay differential equation for the atomic amplitude. We carry out a systematic analysis of the non-Markovian (NM) character of the atomic dynamics in terms of refined, recently developed notions of quantum non-Markovianity such as indivisibility and information back-flow. NM effects are quantified as a function of the round-trip time and phase shift associated with the atom-mirror optical path. We find, in particular, that unless an atom-photon bound state is formed a finite time delay is always required in order for NM effects to be exhibited. This identifies a finite threshold in the parameter space, which separates the Markovian and non-Markovian regimes.


I. INTRODUCTION
The distinction between Markovian and non-Markovian regimes has long been considered a basic one in the study of open system dynamics, i.e., when the system of interest is in contact with an external environment. In qualitative terms, Markovianity is typically associated with the lack of memory effects, a situation which considerably simplifies the theoretical description and yet occurs with good approximation in a number of cases. Assessing the (non-)Markovianity of an open dynamics is a well-understood problem in classical mechanics. In the quantum realm -quite differently -it is not [1][2][3]. Until recently, Markovianity has been almost ubiquitously identified with regimes where the open quantum dynamics is well-described by a master equation (ME) of the Kossakowski-Lindblad form [1] -"Lindbladian" dynamics for brevity-. The latter typically gives rise to purely exponential decays of quantities such as mean energy, po! pulations and coherences. A vast and variegated literature, indeed, has used and in some cases still uses the term "non-Markovian" as a synonym of non-Lindbladian.
Over the last few years, however, a considerable amount of work has been devoted to the refinement of the very notion of non-Markovianity (NM) of a quantum dynamics, with the aim of providing its rigorous identification and quantification [3]. Several new definitions of NM have been proposed, each associated with a specific quantitative measure [4][5][6][7]. A particularly intuitive one is the so called "BLP" measure [6], which identifies NM with the occurrence of quantum information (QI) back-flow between the system and environment (i.e., there exist times at which the latter is able to return QI to the former). To appreciate how these recent studies are affecting the pre-existing paradigm of NM, suffice it to say that certain well-known integro-differential MEs were shown to have zero BLP measure [8], despite for a long time a ME of this sort had been regarded as a typical NM process. The analysis of NM from this re! newed perspective has been recently applied to a number of systems such as atoms in lossy cavities [9,10], spin-boson models [11], spin chains [12] and ultracold atoms [13]. A major motivation to explore different physical scenarios is that studying the emergence of NM in different environmental models helps our understanding of NM itself, a concept whose physical meaning is currently under debate [14].
In this paper we contribute to these efforts by studying NM in the emission process of a quantum emitter or "atom" in front of one mirror, a model that can be solved exactly under very general approximations [16][17][18]. One of the strengths of the considered system -as explained in more detail belowis that the crucial parameters ruling the occurrence of NM effects have a clear physical interpretation. In particular, within the limits of validity of the model, our study clearly illustrates how the non-Markovianity of the atomic emission is affected by imposing simple boundary conditions on the radiation field. In this spirit it is worth recalling that, even in the light of modern NM measures, spontaneous emission of a single atom (in vacuum) embodies the paradigmatic Markovian open dynamics: the emitted radiation simply travels away from the atom, so that the latter has no chance to retrieve information about its past dynamics from the electromagn! etic field (i.e. the environment). A typical way to establish information backflow in this scenario is to impose a geometrical confinement of the field, for example by means of a highfinesse cavity. The NM of an atom in a cavity is often analyzed by assuming an effective Lorentzian-shaped spectral density (SD) centered at a cavity protected frequency [1,9], and in the strong coupling regime can result even in vacuum Rabi oscillations [15], an indisputably non-Markovian phenomenon. A well-known implementation of a cavity is a Fabry-Perot resonator, which features a pair of mirrors. There is no fundamental reason, yet, that prevents NM from taking place even with only one mirror. Rather, in this context, introducing a single mirror in front of an atom appears to be the minimum geometrical confinement enabling the emergence of NM. Thus, from this viewpoint a simple atom-mirror setup -otherwise termed an atom in a half-cavity -can be regarded as a more fundamental system than a cavity to clarify how NM arises in the interaction of matter and geometrically confined light. Specifically, we will focus on a two-level atom where the emitted radiation is constrained to travel along a semi-infinite one-dimensional (1D) waveguide featuring a linear photonic dispersion relation. The finite end of the waveguide behaves as a perfect mirror, forcing part of the emitted light to return |e |g ! 0 lens% lens% quantum%emi,er% trapped%ion% (a) Semi-infinite waveguide, whose only end (behaving as a perfect mirror) lies at x = 0, coupled to a two-level quantum emitter, such as a quantum dot, at x = x 0 . (b) Free-space implementation featuring a trapped super-cool ion (quantum emitter), a real mirror and highnumerical-aperture lenses.
to the atom; one may expect such feedback mechanism to allow for information backflow, hence NM. Also, the finite time taken by a carrier photon to perform a round trip between atom and mirror (time delay t d ) should reasonably behave as an environmental memory time and hence as a key parameter to the occurrence of NM. The restriction to 1D geometry, while certainly a theoretical convenience, also ensures that a significant fraction of emitted light must return to the atom, which intuitively should enhance interference phenomena. These are ruled by the interplay between a phase parameter φ, related to the atom-mirror optical path for a carrier photon, and the dimensionless parameter γt d , that is, the time delay rescaled by the spontaneous emission rate γ. We wonder if and how such interference affects NM. We find it remarkable that the occurrence of NM can be investigated as a function of quantities with a clear physical interpretation. This paper is organized as follows. In Section II, we briefly review the model under investigation, focussing the open system dynamics that the atom undergoes when it emits in vacuum. In Section III, we tackle the problem of employing a reliable criterion to witness NM in the system under study and select a valid NM measure for a rigorous quantification. We explain our choice to use the measure in Ref. [7]. The central findings of this work are then presented in Section IV, where we analize the occurrence of NM effects as a function of the two parameters γt d and φ mentioned above. Special emphasis is given to the characterization of the threshold separating the Markovian and non-Markovian regions in the corresponding parameters space. We summarize our results and deliver some final comments in Section V. Further details on the treatment of the atom-mirror dynamics are given in Appendix A.

II. SHORT REVIEW OF THE MODEL
The model we consider [see Fig. 1(a)] comprises a semiinfinite 1D photonic waveguide lying along the positive xaxis, containing a two-level quantum emitter (atom) placed at x = x 0 . The waveguide termination at x = 0 is assumed to behave as a perfect mirror, imposing a hard-wall boundary condition on the field. Several experimental implementations of the model are possible, involving a variety of quan-tum emitters embedded in several types of waveguides (see e.g. Refs. [21][22][23][24][25][26][27] and Ref. [28] for a more comprehensive list). As shown in Fig. 1(b), a free-space implementation is viable as well along the lines of Ref. [29]. This makes use of a trapped ion, a standard mirror and a pair of high-numericalaperture lenses. We remark that the 1D geometry is an idealization, and that the model could be refined by assuming the presence of external field modes into which the atom can decay [17,18].
The ground and excited states of the atom are denoted by |g and |e respectively, with energy separation ω 0 ( = 1). The waveguide supports a continuum of electromagnetic modes, each with associated wave vector k and frequency ω k . We assume that a linear dispersion relation of the form ω k ω 0 + υ(k − k 0 ), where υ is the photon group velocity and k 0 the carrier wave vector (ω k 0 = ω 0 ), is valid for a sufficiently broad band of modes around the atomic frequency ω 0 .
The atom's emission process was first studied in the 80s [16] and, more recently, revisited and extended in Refs. [17,18]. For the purposes of this work, it will be sufficient to recall the essential results allowing us to describe the reduced dynamics of the atom (the field being initially in the vacuum state). For more details we refer the reader to Appendix A and Refs. [16][17][18]. If the reduced state of the atom in the basis {|e , |g } at time t = 0 is with ρ gg +ρ ee = 1 and ρ eg = ρ * ge , it can be shown that at a later time t its state reads Here, ε(t) is the probability amplitude to find the atom in state |e at time t when ρ 0 = |e e|. In a frame rotating at the atomic frequency ω 0 , the amplitude ε(t) obeys the delay differential equationε Where t d = 2x 0 /υ is the time delay [time taken by a photon to travel from the atom to the waveguide end and back, see Fig. 1(a)] , θ(t) is the Heaviside step function, and the phase φ = 2k 0 x 0 is the optical length for a carrier photon, corresponding to twice the atom-mirror path [in our convention, the π phase shift due to reflection is taken into account by the different signs of the two terms on the righthand side of Eq. (3)]. The crucial assumption in deriving Eq. (3) is that the linearization of the waveguide dispersion relation has to be valid in a band of frequencies broader than the atomic width γ and the inverse of the delay time t −1 d (see Appendix A). Note that this may still allow to have delay times much shorter than the spontaneous emission lifetime, i.e., γt d 1. Eqs. (2) and (3) fully determine the open dynamics of the atom. Note that the first term on the righthand side is associated with standard spontaneous emission. Instead, the feedback introduced by the presence of the mirror is represented by the second term. This is proportional to θ(t − t d ) meaning that, as expected, the atom undergoes standard spontaneous emission up to time t = t d . After this, light emitted in the past can interfere with radiation emitted in the present as well as interact with the atomic dipole moment (i.e., excitation amplitude). Such interference process is witnessed by the phase factor e iφ and, in general, can dramatically affect the dynamics. In particular, it can inhibit the full de-excitation of the atom for φ = 2nπ (n integer), and in the regime γt d 1 it essentially prevents spontaneous emission altogether [18].

III. MEASURING QUANTUM NON-MARKOVIANITY
As discussed in the Introduction, a number of NM measures have been proposed. A known issue is that, in general, such indicators are not equivalent and cases can be found where one of them vanishes while another one does not [3]. A further problem is that their calculation is typically quite involved and may require optimization procedures. Such hurdles, yet, are mostly avoided in our case. The dynamical map in the form described by Eq. (2) can indeed be recognized as an amplitude damping channel. This type of channel for the atomic dynamics also arises in the case of the Jaynes-Cummings model and for an atom coupled to a lossy cavity with a Lorentzian spectral density [1]. In all these cases, a reliable criterion to test occurrence of NM can be expressed as [3,9] d|ε(t)| dt < 0 ∀t > 0 ⇔ the system is Markovian . (5) In equivalent words, if |ε(t)| (in fact, the atomic average energy) grows at some stage of time evolution (even though it may eventually fade away) then the dynamics is non-Markovian (and vice versa). This criterion relies on the demonstrable property [3] that any open dynamics of the form (2) is divisible if and only if d|ε(t)|/dt ≤ 0 at any time, where indivisibility is recognized as a major trait of NM [3]. Moreover, for this type of dynamics, relevant and in general nonequivalent measures of NM -such as those in Refs. [5][6][7] -vanish iff condition (5) holds. In our specific case, using Eq. (3) and the fact that the derivatives of |ε(t)| and |ε(t)| 2 have the same sign, criterion (5) is equivalent to the condition [31] d dt While the study of conditions (5) and (6) is sufficient to distinguish between Markovian and non-Markovian regimes, e.g. for assessing the existence of a threshold in parameter space separating the two, a specific measure has to be chosen in order to quantify NM. In this work, we adopt the recently introduced geometric measure of NM [7]. In general, this is defined as |"| 4 Since |ε(t)| 4 is a monotonic function of |ε(t)|, Eq. (8) enjoys a particularly straightforward connection with criterion (5), making it a natural choice for our purposes. We stress, however, that this is an arbitrary choice since, as anticipated, the qualitative predictions on NM are mostly measureindependent for the present dynamics.

IV. OCCURRENCE OF NON-MARKOVIANITY
Before explicitly computing N, some general expectations on the emergence of NM can be formulated based on Eqs. In the regime where γt d is negligible Eq. (3) can be approximated asε and thus becomes local in time with time-independent coefficients. The corresponding behavior of |ε(t)| and any power of it is clearly an exponential decay, as shown in Fig. 2(a) for |ε(t)| 4 with γt d = 0.05. The dynamics therefore reduces to a Lindbladian one, hence Markovian [17,18]. In such limit, the mirror feedback does not induce any NM, although -depending on φ -it can strongly affect the effective spontaneous emission rate, which can even be arbitrarily small for φ approaching 2nπ, in line with our previous discussion, or double for φ = nπ. Importantly, one has to single out the special case φ = 2nπ, where a bound atom-photon state is formed regardless of the value of γt d [18]. As we show in Section IV C below, the dynamics for φ = 2nπ is NM regardless of γt d .

B. γt d 1: interference-free non-Markovian regime
The opposite regime takes place for γt d 1, which means that the time delay is far longer than the characteristic spontaneous emission time in absence of the mirror. The fraction of light emitted towards the mirror and then reflected back returns to the atom when this has already decayed to the ground state (and the light emitted in the opposite direction has fully departed). Such reflected light is reabsorbed by the atom and then emitted again in either direction and so on. As a consequence, in the regime γt d 1, |ε(t)| 4 exhibits successive spikes of decreasing height as shown in Fig. 2(b). Such behavior occurs independently of φ since, owing to the long retardation time, back-reflected light cannot recombine with light emitted towards the free end of the waveguide and no interference takes place. Criterion (5) thus entails that in this regime the dynamics is certainly non-Markovian. To compute the corresponding N [cf. Eq. (8)] we note that, as discussed in Ref. [17], in the limit γt d 1 ε(t) reduces in each interval to the last non-zero term of sum (4). Therefore, in each time interval [mt d , (m+1)t d ] (m is a positive integer) which is explicitly independent of φ. It is immediate to prove that the time derivative of this function is positive within the subinterval [mt d , mt d +2m/γ], which is in agreement with the behavior in Fig. 2(b). Applying now Eq. (8), we find where the convergence of the series is ensured by Stirling's approximation formula n! n n e −n √ 2πn. Indeed, the summand in Eq. (11) asymptotically approaches (2πm) −2 . A numerical evaluation of the series provides N γt d 1 0.033.

C. General case: intermediate values of γt d
Given the behavior in the limiting cases illustrated above, it is now interesting to investigate whether as γt d grows from zero the system suddenly enters the non-Markovian regime or, instead, there is a finite threshold to trespass. If so, how does this threshold depend on φ? Moreover, we wonder if the degree of NM as given by Eq. (11) is the maximum possible or, instead, N can be higher at lower values of γt d (due to interference effects, we may expect that the answer to this question depends on the phase φ). The regime of intermediate values of γt d features quite a rich physics with a variety of possible behaviors, as can be seen from Figs. 2(c) and (d) for two different values of γt d .
Although exact, the solution (4) of Eq. (3) is unfortunately complicated enough to prevent either N or even the mere NM condition (5) from being worked out in a compact analytical form. We have therefore carried out a numerical computation of N by tabulating |ε(t)| 4 at the nodes of a time-axis mesh. Next, it was checked that the outcomes were stable with respect to the number of mesh points and the length of the overall simulated interval. Fig. 3 shows a contour plot of N as a function of φ and γt d . The considered range of the phase φ is [0, 2π] due to the periodicity of the exponential. To begin our analysis of Fig. 3, we first observe that, as expected, N = 0 if γt d 1 [32] (regime of negligible γt d , see Subsection IV A). On the other hand, as γt d grows (regime of very large γt d ) N converges to N γt d 1 0.03 regardless of φ, in line with the discussion related to Eq. (11). As predicted, such asymp-totic value is independent of φ, which is witnessed by the fact that as γt d grows the profile of N becomes more and more flat with respect to φ. For a set value of γt d , the maximum of N is numerically found at φ = 2nπ and its minimum at φ = (2n + 1)π, where n ≥ 0 is an integer nu! mber. Such values of the phase shift correspond to the atom sitting at a node and antinode, respectively, of the field mode of wave vector k 0 , that is, the mode resonant with the atomic transition [the sine function in Eq. (A1), for this particular mode, can be recast as sin(φ/2)]. This might appear as counter-intuitive since NM is usually expected to increase with the effective atomfield coupling, which in turn grows with the field amplitude at the atom's location. However, it must be kept in mind that considerations about NM are typically model-dependent. For our system, a reasonable interpretation of these results can be given as follows. We note that the parameter φ encodes crucial information about the interference properties of the carrier wave-vector k 0 , around which we expect to find most of the emitted radiation. More specifically, a carrier photon acquires a phase φ+π in a round trip between atom and mirror (the term π due to mirror reflection). Thus, when φ = 2nπ, the reflected field will return to the atom with an overall phase π relative to the radiation that has been emitted towards the free end of the waveguide, resulting in destructive interference between the two. This effectively slows down the emission process, so that part of the emitted light can be expected to remain in the atom-mirror interspace for a significant time, which favours the occurrence of multiple re-absorptions [these bring about NM in virtue of Eq. (5)]. Setting instead φ = nπ (antinode), the interference between the reflected field and the freshly emitted one becomes constructive, thus enhancing the emission of radiation in the direction opposite the mirror. Obviously the latter is unable to re-excite the atom, which results in a reduced NM compared to the former situation. The difference between the two regimes, hence the gap between the maximum and minimum of N (see Fig. 3), becomes negligible as γt d becomes very lar! ge. This can be understood by noting that, in such regime, the photon coming back from the mirror does not encounter any field with which it can interfere (as the atom will have decayed to the ground state well before a round-trip time). Equivalently, one might explain this by interpreting γt d 1 as the regime in which the "bandwidth" γ is large, compared to the characteristic frequency 1/t d : as γ is increased the fraction of light at the carrier wave vector k 0 thus becomes less dominant in determining the behaviour of the emitted light.
A close inspection of Fig. 3 reveals the existence of a finite region on the φ-γt d plane within which the system exhibits a Markovian behavior, i.e., vanishing N. The shape of such Markovianity region can be better appreciated in Fig. 4. For fixed φ, one can find a finite threshold with respect to γt d separating the Markovian and non-Markovian regime. The height of such threshold ranges from γt d = 0 (for φ = 0) to over γt d 1.4 (for φ = π/2). This indicates that, aside from the special point φ = 0, for fixed γ, k 0 and υ the mirror needs to lie far enough from the atom in order for the system to exhibit NM. Hence, when γt d grows from zero the system in general does not enter suddenly the NM region. Loosely speaking, the system can behave in a memoryless fashion even well beyond the Linbladian regime occurring for γt d 0. The appearance of N! M thresholds in parameter space has been demonstrated in a number of systems such as in Ref. [9,10,13]. Interestingly, Fig. 4 shows the occurrence of thresholds even with respect to the phase shift φ for a fixed value γt d (provided that this is lower than the threshold maximum). Owing to the discussed periodicity in φ, this means that a continuous increase of φ makes the system cross in succession interspersed regions of Markovian and non-Markovian behavior. Interestingly, this can be achieved in practice by continuously detuning the atom's frequency (which is routinely attained through local fields) since this is easily seen to be equivalent to a change in φ [18] (provided that the group velocity does not vary significantly in the explored frequency range). Both in terms of maximum amount of NM and threshold height, our results show that the most non-Markovian effects are found for a phase φ = 2nπ. As anticipated, it can be demonstrated [18] that such value of φ enables the formation of an atom-photon bound state in the atom-mirror interspace. This is in line with recent works pointing out the connection between NM and bound system-bath states [33]. In our specific system, it can be shown [18] that an atom-photon bound state of energy ω 0 (hence it is a bound state in the continuum [34]) is formed for φ = 2nπ between the atom and mirror (i.e., the corresponding photon density is identically zero for x > x 0 ). As a consequence, when φ = 2nπ part of the atomic excitation remains trapped according to [18] which corresponds to the overlap between the atom's excited state and such bound state. Note that the trapping is reduced for increasing γt d . On the other hand, from Eq. (3), ε(t d ) = e − γt d 2 which is lower than ε(t→∞) for any γt d > 0. Hence, |ε(t)| must necessarily increase at some time, which in the light of criterion (5) proves that the system is always non-Markovian at this special value of the phase. Adopting a standard viewpoint in the theory of open quantum systems, we further observe that the phase φ determines the position of the atomic frequency with respect to the spectral density (SD) of the "photonic bath". For our model, the spectral density is simply proportional to the square of the atomphoton coupling [1], and can be expressed as where we have defined the atom-photon detuning ∆ ≡ ω−ω 0 and we used the identity υ(k − k 0 ) = ∆. This leads to interpreting t d as the parameter ruling the width of the SD: as t d grows, J(∆) exhibits an increasingly oscillatory behaviour. In the limit t d → 0 and for fixed γ, the SD becomes flat which results in a Lindbladian dynamics (see Subsection IV A and Ref. [20]). At the same time, the behaviour of J(∆) around resonance is decided by φ. Interestingly, such discussion allows for a natural comparison between our atomic dynamics in a single-mirror setup and that occurring in a lossy cavity featuring a Lorentzian SD. In the latter case, the dynamics is characterized by two dimensionless parameters: these are γλ −1 , where λ measures the SD's width, γ being again the spontaneous emission rate in the "flat spectrum" limit, and !δ/γ, with δ the atomic detuning with respect to the cavity protected frequency (at which the maximum of the SD occurs). Significantly, despite major differences between the two systems, also the lossy cavity model exhibits NM thresholds with respect to both the width parameter and detuning [7,9,10]. While a lossy cavity has long been considered the paradigmatic system in which to investigate the emergence of NM effects, we find it interesting that a similarly rich structure can occur even with a single mirror. We also observe that the strength of NM effects, as quantified by Eq. (7), appears comparable in the two models, if the Lorentzian SD is taken in the "bad cavity limit" γλ −1 5. This can be appreciated by comparing Fig. 3 and the results in Ref. [7]. A significant difference between the two models, however, is the fact that the single-mirror setup features an absolute maximum N !max 0.07 as a function of the model parameters, wh! ile for the lossy cavity N can be made arbitrarily large by increasing the cavity quality factor.

V. CONCLUSIONS
We have studied the occurrence of NM in the emission process of an atom coupled to a one-dimensional field, in the presence of a single mirror which imposes a hard-wall boundary condition on the latter. In general, the resulting open dynamics of the atom exhibits a non-exponential behaviour with a rich structure. Adopting the non-divisibility of time evolution as the chosen definition of NM, and a corresponding NM quantifier proposed in Ref. [7], we have studied the strength of NM effects in our system as a function of the two effective parameters characterizing the model: the rescaled round-trip time γt d and the phase φ. While analytical results have been provided in the limiting cases γt d 1 and γt d 1, a numerical approach has been adopted for the intermediate regime γt d ∼ O(1). Remarkably, a finite region in parameter space can be identified where no NM occurs, its boundary defining a NM threshold. For any fixed value of γt d , the maximum NM is found at φ = 2nπ, where a bound atom-photon state is formed. Interestingly, finite Markovian thresholds occur with respect to both the SD width parameter and atomic detuning, a structure which is also exhibited in the open dynamics of an atom in a lossy cavity with Lorentzian spectral density. A deeper and more rigorous insight into the relationship between the NM effects in such cavity model and those occurring in the half-cavity treated here can be gained by introducing a second imperfect mirror in the latter model. The lossy cavity dynamics is then obtained in the limit of negligible time delay! s [16] (a cavity model featuring non-null time delays has been investigated in Ref. [35,36]). The analysis of NM for such a two-mirror model, which can be regarded as an ab initio -instead of phenomenologicaldescription of a lossy cavity, is currently under investigation [37].
where ε(0) = 1 and ϕ(k, t) is the field amplitude in k-space. From this, it is immediate to derive Eq. (2) for the evolution of the atomic reduced state. To work out ε(t), one makes use of the the time-dependent Schrödinger equation i|Ψ(t) = H|Ψ(t) , which results in a system of differential equations for ε(t) and ϕ(k, t) [17,18]. Two approximations are then made. (i) It is assumed that the photon dispersion relation can be linearized as ω k ω 0 +υ(k−k 0 ), where υ = dω/dk| k=k 0 is the photon group velocity and k 0 is the wave vector corresponding to the atomic frequency, i.e., ω κ 0 = ω 0 . (ii) The integral bounds are approximated as k c 0 dk ∞ −∞ dk. These routine approximations [20], together with the RWA mentioned earlier, rely on the fact that only a narrow range of wave vectors around k=k 0 is expected to give a significant contribution to the dynamics. Since the range ! of frequencies involved in the the atomic dynamics is ruled by the two parameters γ and t d , we deduce that the linearization of the waveguide dispersion relation is a good approximation in a band of frequencies broader than the atomic width γ as well as the inverse of the delay time t −1 d . Clearly, a further requirement is that the time delay t d should be much larger than the optical period ω −1 0 , in order to avoid the breakdown of the RWA. In specific implementations of the model, these assumptions have to be checked a posteriory for consistency. Once we set in a rotating reference frame such that ε(t) → ε(t)e −iω 0 t , ϕ(k, t) → ϕ(k, t)e −iω 0 t and the field variables are expressed in terms of the atomic excitation amplitude [17,18], we end up with Eq. (3) in the main text.